Manuscript IX-B-ST@1 · IX-B-ST.tex · theorem · f6af1523

Theorem: One destructive clock instrument

Finite numerical certificates and a destructive clock › Complete destructive clock · lines 1639–1667 · proof included or linked

IX-B-ST@1#thm:instrument, IX-B-ST@1#eq:clock

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The calibrated clock readout is

M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})] .

It is CPTP, including after tensoring with $id_{E}$ , and

Tr_{R} M_{C} (X) = Tr_{C} X .

It returns subnormalized conditional blocks and consumes the quantum clock names. The geometric coordinates and metric are not consumed.

Proof. The recorded Kraus operators are $∣ k ⟩_{R} \otimes D_{k}$ . Their squared moduli sum to $I_{C S}$ . Expanding $X$ in the calibration basis proves the partial-trace expression and the record-discard identity. Tensoring these identities with $I_{E}$ proves the complete statements. The declared destructive output interface has no quantum $C$ -factor, whereas the host is fixed metadata.

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IX-B-ST@1#eq:clock · line 1642
$M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})] .$
unlabeled · line 1650
$Tr_{R} M_{C} (X) = Tr_{C} X .$

Theorem: One destructive clock instrument

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